Twin index

From Online Dictionary of Crystallography

Contents

Definition

A twin operation overlaps both the direct and reciprocal lattices of the individuals that form a twin; consequently, the nodes of the individual lattices are overlapped (restored) to some extent (twinning). The reciprocal n of the fraction 1/n of (quasi)restored nodes is called twin index

Let (hkl) be the twin plane and [uvw] the lattice direction (quasi)-normal to it. alternatively, let [uvw] be the twin axis and (hkl) the lattice plane (quasi)-normal to it. For twofold operations (180º rotations or reflections) the twin index is:

n = X/f, X = |uh+vk+wl|

where f depends on the lattice type and on the parities of X, h, k, l, u, v and w, as in the following table

Lattice type

condition on hkl

condition on uvw

condition on X

n

P

none

none

X odd

n = X

X even

n = X/2

C

h+k odd

none

none

n = X

h+k even

u+v and w not both even

X odd

n = X

X even

n = X/2

u+v and w both even

X/2 odd

n = X/2

X/2 even

n = X/4

B

h+l odd

none

none

n = X

h+l even

u+w and v not both even

X odd

n = X

X even

n = X/2

u+w and v both even

X/2 odd

n = X/2

X/2 even

n = X/4

A

k+l odd

none

none

n = X

k+l even

v+w and u not both even

X odd

n = X

X even

n = X/2

v+w and u both even

X/2 odd

n = X/2

X/2 even

n = X/4

I

h+k+l odd

none

none

n = X

h+k+l even

u, v and w not all odd

X odd

n = X

X even

n = X/2

u, v and w all odd

X/2 odd

n = X/2

X/2 even

n = X/4

F

none

u+v+w odd

none

n = X

h, k, l not all odd

u+v+w even

X odd

n = X

X even

n = X/2

h, k, l all odd

u+v+w even

X/2 odd

n = X/2

X/2 even

n = X/4

When the twin operation is a rotation of higher degree about [uvw], in general the rotational symmetry of the two-dimensional mesh in the (hkl) plane does no longer coincide with that of the twin operation. The degree of restoration of lattice nodes must now take into account the two-dimensional coincidence index Ξ for a plane of the family (hkl), which defines a super mesh in the twin lattice. Moreover, such a super mesh may exist in ξ planes out of N, depending on where is located the intersection of the [uvw] twin axis with the plane. The twin index n is finally given by:

n = NΞ/ξ

References

Chapter 3.1.9 in International Tables for X-Ray Crystallography (1959)